Methods and systems disclosed herein relate generally to adjusting values at the interface between numerical models, and in particular to adjusting prognostic variables between models of different resolutions and dynamics. Modeling submesoscale dynamics, scales of 10 m to 1 km, requires resolutions that preclude global modeling and dynamics that are unnecessary for global scales. In order to model submesoscale dynamics as part of a global forecasting system, as is the ultimate goal, a smaller model domain with open boundaries derived from a coarser, larger scale model is required. Method(s) may be needed to handle the open boundaries of the submesoscale domain. As the name open boundary implies, fluid must be able to flow freely in and out of the model domain. Also, scales of motion too large to be contained in the submesoscale domain must correctly influence the flow in the domain. The boundaries of the submesoscale domain should be truly transparent, i.e. they should be invisible in the results of the modeling.
For a one-way coupled case, open boundary conditions between a submeso scale, nonhydrostatic inner domain and a mesoscale, hydrostatic outer domain may be required to meet certain conditions. The submesoscale resolution allows dynamic features to be resolved which may not be captured at mesoscale resolution. Furthermore, in order to model submesoscale dynamics which includes nonlinear internal waves (NLIWs) with large amplitude (100 s of meters) and subinertial eddies and filaments, a nonhydrostatic model with resolution of 10's to 100's of meters in the horizontal and meters to 10's of meters in the vertical may be used. The difference in the physics between the nonhydrostatic model and the hydrostatic model can lead to differences in the solutions that exacerbate the problems of interfacing the nested models. When a mismatch of the solutions occurs at the interface between the domains, spurious reflections and erroneous perimeter currents along the boundaries can result. Also, incorrect geostrophic currents can be set up along the open boundaries by erroneous density gradients across the boundaries.
Open boundary conditions, (OBCs) can transparently provide information to the inner domain from the outer domain and propagate information out of the inner domain. In one-way coupling, the information propagated out of the inner domain does not affect the outer domain. Because of the differences in physics and resolution, there could be inherent differences between the mesoscale solution and the submesoscale solution. Prognostic variables from the coarse grid can be interpolated onto the fine grid in a manner that conserves basic properties and scales. This interpolation can be problematic since the most common interpolation schemes are not conservative, e.g. bilinear and nearest neighbor. Methods for formulating and applying OBCs to limited area models can be divided into two categories: 1) adaptive and 2) consistent.
Adaptive methods differentiate between incoming and outgoing fluxes and apply different OBC values based on the direction of the flux. Consistent OBCs apply the same values regardless of the direction of the boundary flux. The advantage of using adaptive OBCs is that different OBCs can be applied for each case. This allows the response to be tailored to the characteristics and scales of the outer domain or inner domain solution as appropriate. Disadvantages of adaptive OBCs include: determining which points qualify as inflow and which qualify as outflow, handling transitions between adjacent inflow and outflow areas, and handling strong tangential flows. Adaptive OBCs are further complicated by the fact that any open boundary point could simultaneously be an inflow point for some scales and an outflow point for others.
Consistent OBCs apply the same method to all open boundary points. These methods are typically a form of wave radiation, absorption or relaxation scheme. Problems with radiation schemes stem from the fact that multiple waves with different phase speeds can exist in realistic cases, whereas the radiation schemes can use only one phase speed. Absorption schemes typically have a region where the viscosity is artificially increased to damp the solution near the boundaries. Relaxation schemes can involve a region near the boundaries in which the inner and outer values are smoothly merged. One issue with relaxation and absorbing layers is that the equations governing the motion in the boundary layer are not the correct equations for any approximation of the physics so spurious solutions can result. For simulations involving idealized or climatological case studies of processes, the large scale flow can be prescribed as constant or a simple function of time and the OBCs merge these values with the flow field from the submesoscale domain. For predictions whether in real-time, i.e. forecasts, or offline, i.e. hindcasts, the problem is more complicated. In these cases, the OBCs must be able to merge the multiple time and space scales of the larger scale dynamics and the multiple time and space scales of the resolved submesoscale dynamics. What is needed is a method for use of consistent OBCs that overcomes the limitations of currently-used methods. What is further needed is an interpolation method for transporting momentum across a boundary, and overcomes the problems of 1) a secular trend that develops in the numerical solution to the dynamics in a nonhydrostatic domain nested inside a hydrostatic domain, and 2) energy in the form of waves and vorticies that becomes trapped in the inner domain because it is erroneously reflected at the lateral boundaries. What is further needed is a system that creates a set of momentum conserving boundary conditions for the incoming flow and a flow relaxation region to match the inner and outer solutions and prevent reflection into the inner domain